https://file.aiquickdraw.com/mathbot/file/1726582114259ub0thz7c.jpg

The problem asks to find the other angle and sides of a right triangle \( ABC \) where:

- \( \angle C = 90^\circ \) (since it's a right triangle),
- \( c = 25 \) (hypotenuse),
- \( b = 3 \) (one of the legs).

We need to find:

1. The other leg \( a \).
2. The angles \( \angle A \) and \( \angle B \).

### Step 1: Finding side \( a \) using the Pythagorean theorem

For a right triangle, the Pythagorean theorem states:
\[
a^2 + b^2 = c^2
\]

Substitute the values of \( b \) and \( c \):

\[
a^2 + 3^2 = 25^2
\]

\[
a^2 + 9 = 625
\]

\[
a^2 = 625 - 9 = 616
\]

\[
a = \sqrt{616} \approx 24.8
\]

So, the length of side \( a \) is approximately \( 24.8 \).

### Step 2: Finding angle \( \angle A \)

Use the sine function for \( \angle A \):
\[
\sin A = \frac{b}{c} = \frac{3}{25}
\]

Now, calculate \( A \):

\[
A = \arcsin\left(\frac{3}{25}\right) \approx 6.89^\circ
\]

### Step 3: Finding angle \( \angle B \)

Since the sum of angles in a triangle is \( 180^\circ \), and we already know \( \angle C = 90^\circ \) and \( \angle A \approx 6.89^\circ \), we can find \( \angle B \) as:

\[
\angle B = 90^\circ - \angle A = 90^\circ - 6.89^\circ = 83.11^\circ
\]

### Final Answer:

- \( a \approx 24.8 \)
- \( \angle A \approx 6.89^\circ \)
- \( \angle B \approx 83.11^\circ \)

Would you like further details or have any questions?

Here are some related questions:
1. How does the Pythagorean theorem work for other right triangle problems?
2. What are some other methods to calculate angles in a triangle?
3. How would the problem change if the value of \( b \) were different?
4. What is the relationship between sine, cosine, and tangent in a right triangle?
5. Can the Pythagorean theorem be applied in non-right triangles?

**Tip:** Always verify if you're working with a right triangle to decide whether or not the Pythagorean theorem applies.

Find the other angle and sides of the following right triangle (ABC) with C = 90° and c = 25, b = 3.

Learn how to solve a right triangle where C = 90°, the hypotenuse c = 25, and one leg b = 3. Using the Pythagorean theorem, find the length of the other leg, angle A, and angle B in this step-by-step trigonometry problem. Suitable for Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation